# Write the complex number in trigonometric (polar) form

• May 6th 2013, 09:20 PM
boltage619
Write the complex number in trigonometric (polar) form
Write the complex number 3-i in trigonometric (polar) form, with theta in the interval [0 degrees, 360 degrees). Round to the nearest tenth.
• May 6th 2013, 10:04 PM
MarkFL
Re: Write the complex number in trigonometric (polar) form
You want:

$\displaystyle 3-i=r\,\text{cis}(\theta)$

From your other topic where we went from polar to rectangular form, you should be able to see that:

(1) $\displaystyle r\cos(\theta)=3$

(2) $\displaystyle r\sin(\theta)=-1$

and by Pythagoras we know:

(3) $\displaystyle r^2\sin^2(\theta)+r^2\cos^2(\theta)=r^2$

What do we get if we divide (2) by (1)? What quadrant is the angle $\displaystyle \theta$ in?

What do we get if we use (1) and (2) to substitute into (3)?
• May 6th 2013, 10:10 PM
boltage619
Re: Write the complex number in trigonometric (polar) form
If we divide (2) by (1) we get -.333 repeated and im not sure how to substitute 1 and 2 into 3 because my teacher didn't teach us this part...
• May 6th 2013, 10:19 PM
MarkFL
Re: Write the complex number in trigonometric (polar) form
What I mean by dividing (2) by (1) is you should get:

$\displaystyle \frac{r\sin(\theta)}{r\cos(\theta)}=\frac{-1}{3}$

which simplifies to:

$\displaystyle \tan(\theta)=-\frac{1}{3}$

Now, when we use the inverse tangent function, we need to be mindful of which quadrant the angle is actually in. Which quadrant is the sine function negative and the cosine function positive for a given angle? Your calculator is going to give you:

$\displaystyle -90^{\circ}<\theta<90^{\circ}$

and care must be taken to use this, and the fact that $\displaystyle \tan\left(\theta+k\cdot180^{\circ} \right)=\tan(\theta)$ where $\displaystyle k\in\mathbb{Z}$ ($\displaystyle k$ is an integer) to put the angle in the desired form.

So first, what does your calculator tell you the angle is?